The Scientific World Journal

Volume 2013 (2013), Article ID 685621, 9 pages

http://dx.doi.org/10.1155/2013/685621

Research Article

Oscillation of a Class of Fractional Differential Equations with Damping Term

Huizeng Qin and Bin Zheng

School of Science, Shandong University of Technology, Zibo, Shandong 255049, China

Received 19 May 2013; Accepted 11 July 2013

Academic Editors: J. Banaś and M. M. Cavalcanti

Copyright © 2013 Huizeng Qin and Bin Zheng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We investigate the oscillation of a class of fractional differential equations with damping term. Based on a certain variable transformation, the fractional differential equations are converted into another differential equations of integer order with respect to the new variable. Then, using Riccati transformation, inequality, and integration average technique, some new oscillatory criteria for the equations are established. As for applications, oscillation for two certain fractional differential equations with damping term is investigated by the use of the presented results.

1. Introduction

In the investigations of qualitative properties for differential equations, research of oscillation has gained much attention by many authors in the last few decades (e.g., see [1–16]). In these investigations, we notice that very little attention is paid to oscillation of fractional differential equations.

In [17], Jumarie proposed a definition for fractional derivative which is known as the modified Riemann-Liouville derivative in the literature. Since then, many authors have investigated various applications of the modified Riemann-Liouville derivative (e.g., see [18–21]) including various fractional calculus formulae, the fractional variational iteration method, the Bäcklund transformation method, and the fractional subequation method for soling fractional partial differential equations. In this paper, based on the modified Riemann-Liouville derivative, we are concerned with oscillation of a class of fractional differential equations with damping term as follows:

where denotes the modified Riemann-Liouville derivative with respect to the variable , the function , , and denotes continuous derivative of order .

The definition and some important properties for the modified Riemann-Liouville derivative of order are listed as follows (see also in [20–24]):

As usual, a solution of (1) is called oscillatory if it has arbitrarily large zeros; otherwise, it is called nonoscillatory. Equation (1) is called oscillatory if all its solutions are oscillatory.

We organize the next as follows. In Section 2, using Riccati transformation, inequality, and integration average technique, we establish some new oscillatory criteria for (1), while we present some examples for them in Section 3.

2. Oscillatory Criteria for (1)

In the following, we denote , , , , , , , , , , , , . Let satisfy

has continuous partial derivatives and on such that

Lemma 1. Assume is an eventually positive solution of (1), and

Then, there exists a sufficiently large such that on and either on or .

Proof. Let , where . Then, by use of (3), we obtain , and furthermore, by use of the first equality in (5), we have

Similarly, we have , . So, (1) can be transformed into the following form:

Since is an eventually positive solution of (1), then is an eventually positive solution of (12), and there exists such that on . Furthermore, we have

Then, is strictly decreasing on , and thus is eventually of one sign. We claim on , where is sufficiently large. Otherwise, assume that there exists a sufficiently large such that on . Then, is strictly decreasing on , and we have

By (8), we have . So there exists a sufficiently large with such that , . Furthermore,

By (9), we deduce that , which contradicts the fact that is an eventually positive solution of (9). So, on , and on . Thus, is eventually of one sign. Now we assume , for some sufficiently large . Since , furthermore we have . We claim . Otherwise, assume . Then on , and, for , by (12) we have

Substituting with in the previous inequality, an integration with respect to from to yields

which means

Substituting with in (18), an integration for (18) with respect to from to yields

that is,

Substituting with in (20), an integration for (20) with respect to from to yields

By (10), one can see , which causes a contradiction. So, the proof is complete.

Lemma 2. Assume that is an eventually positive solution of (1) such that

on , where is sufficiently large. Then, for , we have

Proof. By (13), we obtain that is strictly decreasing on . So,

that is

which admits (23). On the other hand, we have

which can be rewritten as (24). So the proof is complete.

Lemma 3 (see [25, Theorem 41]). Assume that and are nonnegative real numbers. Then,

for all .

Theorem 4. Assume that (8)–(10) hold. If there exists such that for any sufficiently large , there exist , , with satisfying

where , ,  ; then, (1) is oscillatory or satisfies .

Proof. Assume that (1) has a nonoscillatory solution on . Without loss of generality, we may assume that on , where is sufficiently large. By Lemma 1, we have , where is sufficiently large, and either on or . Now we assume on . Define the generalized Riccati function:

Then, for , we have

Using and (23), we obtain

Let . Then , and . So (32) is transformed into the following form:

Choose , , arbitrarily in with . Substituting with , multiplying both sides of (33) by , and integrating it with respect to from to for , we get that

Dividing both sides of the inequality (34) by and letting , we obtain

On the other hand, substituting with , multiplying both sides of (33) by , and integrating it with respect to from to for , we get that

Dividing both sides of the inequality (36) by and letting , we obtain

A combination of (35) and (37) yields

which contradicts (29). So, the proof is complete.

Theorem 5. Under the conditions of Theorem 4, if for any sufficiently large ,

then (1) is oscillatory.

Proof. For any , let . In (39), we choose . Then, there exists such that

In (40), we choose . Then there exists such that

Combining (41) and (42), we obtain (29). The conclusion thus comes from Theorem 4, and the proof is complete.

In Theorems 4 and 5, if we choose , , where is a constant, then we obtain the following two corollaries.

Corollary 6. Under the conditions of Theorem 4, if for any sufficiently large , there exist , , with satisfying

then (1) is oscillatory.

Corollary 7. Under the conditions of Theorem 5, if for any sufficiently large ,

then (1) is oscillatory.

Theorem 8. Assume (8)–(10) hold, and

where is defined as in Theorem 4. Then every solution of (1) is oscillatory or satisfies .

Proof. Assume (1) has a nonoscillatory solution on . Without loss of generality, we may assume on , where is sufficiently large. By Lemma 1, we have , , where is sufficiently large, and either on or . Now we assume that on . Let , be defined as in Theorem 4. Then we obtain (33), and furthermore,

Substituting with in (46) and integrating (46) with respect to from to yield

which contradicts (45). So, the proof is complete.

Theorem 9. Assume (8)–(10) hold, and there exists a function such that , for , , for , and has a nonpositive continuous partial derivative . If

where is defined as in Theorem 4, then every solution of (1) is oscillatory or satisfies .

Proof. Assume (1) has a nonoscillatory solution on . Without loss of generality, we may assume on , where is sufficiently large. By Lemma 1, we have , , where is sufficiently large, and either on or . Now we assume on . Let , be defined as in Theorem 4. By (46), we have

Substituting with in (49), multiplying both sides by , and then integrating both sides of (49) with respect to from to yield

Then,

So,

which contradicts (48). So the proof is complete.

3. Applications of the Results

Example 10. Consider the following fractional differential equation:

In (1), if we set , , , , , , then we obtain (53). So , , , , . Furthermore, , which implies . On the other hand, , which implies , and then (8) holds. So, there exists a sufficiently large such that on . In (9),

In (10),

In (48), letting , we obtain

Therefore, (53) is oscillatory by Theorem 8.

Example 11. Consider the following fractional differential equation:

In (1), if we set , , , , , , then we obtain (57). So , , , , . Furthermore, , which implies . On the other hand, , which implies . So, there exists a sufficiently large such that on .

From the analysis above, one can see the (8) holds. We now test (9) and (10). In (9),

In (10),

So, (9) and (10) hold. On the other hand, in (44), after putting , , for any sufficiently large , we have

So (44) holds, and then by Corollary 7 we deduce that (57) is oscillatory.

References

S. R. Grace, J. R. Graef, and M. A. El-Beltagy, “On the oscillation of third order neutral delay dynamic equations on time scales,” Computers and Mathematics with Applications, vol. 63, no. 4, pp. 775–782, 2012.

Y. Sun, Z. Han, Y. Sun, and Y. Pan, “Oscillation theorems for certain third order nonlinear delay dynamic equations on time scales,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 75, pp. 1–14, 2011.

L. Erbe, T. S. Hassan, and A. Peterson, “Oscillation of third order functional dynamic equations with mixed arguments on time scales,” Journal of Applied Mathematics and Computing, vol. 34, no. 1-2, pp. 353–371, 2010.

T. S. Hassan, “Oscillation of third order nonlinear delay dynamic equations on time scales,” Mathematical and Computer Modelling, vol. 49, no. 7-8, pp. 1573–1586, 2009.

D. Cakmak and A. Tiryaki, “Oscillation criteria for certain forced second-order nonlinear differential equations,” Applied Mathematics Letters, vol. 17, no. 3, pp. 275–279, 2004.

Z. Han, T. Li, S. Sun, and C. Zhang, “Oscillation behavior of third-order neutral Emden-Fowler delay dynamic equations on time scales,” Advances in Difference Equations, vol. 2010, Article ID 586312, 2010.

M. Bohner, “Some oscillation criteria for first order delay dynamic equations,” Far East Journal of Applied Mathematics, vol. 18, no. 3, pp. 289–304, 2005.

R. P. Agarwal, M. Bohner, and S. H. Saker, “Oscillation of second order delay dynamic equations,” Canadian Applied Mathematics Quarterly, vol. 13, pp. 1–18, 2005.

R. P. Agarwal and S. R. Grace, “Oscillation of certain third-order difference equations,” Computers and Mathematics with Applications, vol. 42, no. 3–5, pp. 379–384, 2001.

N. Parhi, “Oscillation and non-oscillation of solutions of second order difference equations involving generalized difference,” Applied Mathematics and Computation, vol. 218, pp. 458–468, 2011.

S. H. Saker, “Oscillation of nonlinear dynamic equations on time scales,” Applied Mathematics and Computation, vol. 148, no. 1, pp. 81–91, 2004.

F. Meng and Y. Huang, “Interval oscillation criteria for a forced second-order nonlinear differential equations with damping,” Applied Mathematics and Computation, vol. 218, no. 5, pp. 1857–1861, 2011.

D. X. Chen, “Oscillation criteria of fractional differential equations,” Advances in Difference Equations, vol. 2012, article 33, 2012.

D. X. Chen, “Oscillatory behavior of a class of fractional differential equations with damping,” UPB Scientific Bulletin, Series A, vol. 75, no. 1, pp. 107–118, 2013.

B. Zheng, “Oscillation for a class of nonlinear fractional differential equations with damping term,” Journal of Advanced Mathematical Studies, vol. 6, no. 1, pp. 107–115, 2013.

A. Aghili and M. R. Masomi, “Integral transform method for solving time fractional systems and fractional heat equation,” Bulletin of Parana's Mathematical Society, vol. 32, pp. 305–322, 2014.

G. Jumarie, “Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results,” Computers and Mathematics with Applications, vol. 51, no. 9-10, pp. 1367–1376, 2006.

G. Jumarie, “Table of some basic fractional calculus formulae derived from a modified Riemann-Liouville derivative for non-differentiable functions,” Applied Mathematics Letters, vol. 22, no. 3, pp. 378–385, 2009.

N. Faraz, Y. Khan, H. Jafari, A. Yildirim, and M. Madani, “Fractional variational iteration method via modified Riemann-Liouville derivative,” Journal of King Saud University, vol. 23, no. 4, pp. 413–417, 2011.

B. Lu, “Backlund transformation of fractional Riccati equation and its applications to nonlinear fractional partial differential equations,” Physics Letters A, vol. 376, pp. 2045–2048, 2012.

S. Zhang and H.-Q. Zhang, “Fractional sub-equation method and its applications to nonlinear fractional PDEs,” Physics Letters Section A, vol. 375, no. 7, pp. 1069–1073, 2011.

S. M. Guo, L. Q. Mei, Y. Li, and Y. F. Sun, “The improved fractional sub-equation method and its applications to the space-time fractional differential equations in fluid mechanics,” Physics Letters A, vol. 376, pp. 407–411, 2012.

B. Zheng, “(G'/G)-expansion method for solving fractional partial differential equations in the theory of mathematical physics,” Communications in Theoretical Physics, vol. 58, pp. 623–630, 2012.

B. Lu, “The first integral method for some time fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 395, pp. 684–693, 2012.

G. H. Hardy, J. E. Littlewood, and G. P. Pólya, Inequalities, Cambridge University Press, Cambridge, UK, 2nd edition, 1988.